On the convergence rate of a recursively defined sequence

Consider the following recursively defined sequence: $tau _1 = 1,sumlimits_{j = 1}^n {frac{1} {{sumnolimits_{s = j}^n {tau _s } }}} = 1forn geqslant 2, $ , which originates from a heat conduction problem first studied by Myshkis (1997). Chang, Chow, and Wang (2003) proved that $tau _n = log n + O(1) for large n.$ . In this note, we refine this result to $tau _n = log n + gamma + Oleft( {frac{1} {{log n}}} right). $ . where γ is the Euler constant.